Fractional
iteration of the function f(x) = 1/(1+x)

1) The problem

In this article I deal with the fractional
iteration of a very simple function. We use:

f(x)
= 1 / (1 + x)

and in the iterational notation:

f°0(x) = x f°1(x) = f(x) f°h+1(x) = f°h(f°1(x))

The integer iteration for h=0..inf produces the sequence of
continued fractions

(where the bracketed expression denote the
infinite continued fraction)

which converges to the golden ratio ~0.618…
for all x.

It is not obvious, how to arrive at a
formulation which allows interpolation to fractional iterates. We'll see, that
such a formulation can be given when the generalization of fibonacci-numbers to
fractional and general indexes is introduced, see chap 3. First I use the
standard-procedure: rewrite the function as powerseries in x, generate the matrix-operator F; if the powerseries f(x) has a
constant term, recenter around a fixpoint t:g(x) = f(x+t)–t , compute the triangular
matrix-operator G (the Bell-matrix for functon g(x)) and
perform regular iteration by powers of G,
where fractional iterates are well defned by fractional powers of G and that fractional powers can be computed
using diagonalization:

G
= W *D * W-1 Gh = W * Dh * W-1

where the fractional and general power of the
diagoal-matrix D can be implemented by the
general power of its scalar diagonal-elements:

Dh = diagk=0..inf(dk,kh)

2) Handling by expansion into powerseries and
diagonalization

2.1) f(x) expressed as powerseries

First express f(x) as powerseries

f(x)
= 1 – x + x2 – x3 + … – …

which is the well known expansion.

Caveats: Looking precisely at it, this substitution must be justified; let's
consider at least he following:

·While for the powerseries-expansion of
f(x)
we know that it must be |x| < 1 for the series to converge, there is
no such restriction in the fractional notation. Well, we know that the series
can be analytically continued to match the values of the fractional notation –
but this indicates that we need some care when we replace one notation by another
one.

·Another discrepancy is the different
handling of the domain for the fractional representation vs. the powerseries
representation. Under the iteration-aspect this is a more serious aspect; it is
for the fractional notation, that the domain thins out the higher the iteration
of the fraction is assumed. The domain for 1/(1+x) is IC\{-1},
the domain for 1/(1+1/1+x))
is IC\{-1,-2}, and this continues for each iteration.

Thus: if we want to
discuss the iteration of 1/(1+x) depending on the parameter
"height" the domain-restriction is somehow part of this
function-parameter. I don't have an idea, how this must be discussed for
fractional or continuous iteration – maybe the domain is completely thinned
out… . However, for the series-representation we don't see such a distinctive
characteristic, we only know, that for all |x|>= 1 we have no convergence and must use
analytical continuation to allow to extend the domain.

I do not want to go
deeper in this here; I assume these problems are solvable and proceed in discussing
the properties of the iteration of the powerseries.

2.2) Recentering of f(x): fixpoint-shift
to g(x)

The above powerseries f(x) has a constant term and is
thus not well suited for fractional iteration. But this may be resolved by
reformulation using fixpoint-shift.

The function has two fixpoints x0 and x1which
can easily be derived from the fractional representation x = 1/(1+x).

Integer and fractional powers of that matrix
will then contain the coefficients for the accordingly iterated function g°h(x) in its second column. The
fractional powers are computed by diagonalization and fractional powers of the
matrix-diagonal. Since G is
triangular we have finite polynomials in powers of φ only, so we have exact
expressions in terms of φ for each coefficient of the powerseries
for the fractional iterates.

2.5) Diagonalization: G = W *
D * W-1

Diagonalization means, that the matrix G is decomposed in three matrices, where W and W-1 are lower triangular and D is
diagonal. For triangular Bell-matrices like in this case, where the elements of
the diagonal are not all powers of 1, this can always be done by a relatively
simple procedure, which gives exact coefficients in all tree matrix-factors.

The matrices W, D
and W-1
are (top left segments, matrices have infinite size)

diag()

Exact, where @ is used for 1/sqrt(5)

Exact, where @ is used for φ

Exact, where @ is used for 1/sqrt(5)

diag()

By this construction, W
is simply a power of the binomial-matrix ; it is

W
= P –1/sqrt(5) W-1 = P 1/sqrt(5)

Using the rules of diagonalization we can now
express powers of G by powers of the
diagonalmatrix of eigenvalues D:

Gh = W * Dh * W –1

and precisely we have to write

Gh = W * diag(r=0..inf ( dhr)) * W-1

where dh =(-(φ²))h must be determined first and of
this for each entry in Dh the appropriate integer power is then needed. This care is not
needed for iteration to integer heights, but will be needed when continuous
iteration with fractional or complex h is attempted.

the latter functional description using the
second column of (the current power of) G.

The same formula serves for general powers of G as discussed in the previous paragraph:

V(x)
* Gh = V( g°h(x) )

and introducing the fixpoint-shift:

V(x–x0)
* Gh = V( f°h(x)–x0 )

2.7) Use of the Schröder-function

Using the diagonalization we have implemented
the fractional iteration. But also the matrices W
and W-1
can be seen as the operators for the Schröder-function σ and its inverse σ°-1 for
the function g(x):

g°h(x) = σ°-1( d h
*σ (x))
and f°h(x) = σ°-1( d h
*σ (x–x0)) + x0

Here the cofficients of the Schröder-function
are in the second column of W
resp. W-1 and the coefficient dh is the h'th power of
the second entry in the diagonal D:

Since the Schröder-funtions are again geometric
series, we can give closed forms for them:

σ
(x) =x sqrt(5) /( sqrt(5) + x ) σ°-1(x) = x sqrt(5)/( sqrt(5) –
x)

and can determine exact coefficients in terms
of powers of φ .

2.8) Observations/Questions

We noticed, that the second eigenvalue of the
matrix G is negative; so fractional
powers require complex values for the diagonal, and thus, by this method, we
get complex values for real fractional iterates of real initial values.

I'd like to see possibly concurring methods to
compare the results here.

Below are some plots for fractional iterates
for h=0 to 2
in steps of 1/10
for the initial values x=1 and x=0.25.

Questions:

·is there a concuring method for
fractional iteration? What values does it produce

·how can we express the function using
the second fixpoint for fixpoint-shift according to the above rationale

Gottfried Helms; 28.01.2009

Update 10'2009: I could give an aswer to the
first question myself. See next chapter.

3) Handling by interpolation of functional iterates

If we look at the coefficients of the fraction
when integer-iterated, we notice the progression related to the
fibonacci-numbers.

which is not difficult to show in generality by
induction. Now there is a continuous extension for the fibonacci-numbers, such
that we can use fibonacci-numbers of fractional and finally of arbitrary complex
index. The formula for a function, containing the height-parameter h as well reads

where

and

Writing for a constant argument x (for instance
x=1) a new function depending only on h:

fh(h)
= f°h(1)

we get the same values and the same plot as in
the previous chapter. See appendices.

Gottfried Helms; 5.10.2009

4) Loose ends

4.1) f(f(x))
as smoothed function, which has real fractional iterates

Since the second eigenvalue of G is negative, iterates of f(x)
produce this spiraling behaviour when iterated in noninteger stepwidth (see
plot "pic 1").

If we use ff(x) = f(f(x)) as iterable function, then its
second eigenvalue is positive and there should be a fractional iteration giving
a real trajectory only.

We have for ff(x) and the inverse ff°-1(x) the definitions:

Since ff(x) has a positive eigenvalue, we
can find a half-iterate which has also a positive eigenvalue, again using
fixpoint-shift by φ. Call the fixpoint-shifted version of ff(x)gg(x), such that ff(x) = gg(x+ φ )– φ . Call the half-iterate of gg(x)gg°0.5(x), where I found the analytic
description of the coefficients heuristically (and they are not proven
correct). Assume the correctness, then for the half-iterate of ff(x) we get:

finally

(1)

which provides meaningful results, if we, for
instance start with x=1, compute x1=ff°0.5(x) , x2=ff°0.5(x1) and see that the
results match: x2 = ff(x)
. Also the taylor-coefficients of ff(x) and ff°0.5(ff°0.5(x)) come out to be equal as far as
checkable by Pari/Gp, and
finally, using the symbolic algebra-system Maxima,
we simplify using the definition (1):

ff°0.5(ff°0.5(x)) = (1+x)/(2+x)

which is the expected result

4.2) ff(x) as
iteration-series

If I want to express a function f(x) as
a series of iterates of a function ft(x) (let's call this an "iteration-series", I used
"powertowerseries" in context of tetration here), then we have the
following relation for a possible definition of ft(x):

The trajectory of iterates of ft(x) is
very chaotic and seems to not to converge in a classical sense.

Fixpoints:

The fixpoints are x0=0, x1,x2=-2
, one quadratic singularity at x=-1

(see a plot at pic 5 below)

5.) Plots

Pic 1:

Trajectories for fractional iteration f°h(x) for two initial values x:

Pic 2:

The vectorial angles between coordinates of
iterates and the coordinate of the limit [φ,0*î] taken as origin are nearly equal,
the angle is nearly a linear function of the iteration-height h

Pic 3:

The complex coordinates on the trajectory for x=1 can
be translated moving the origin to the limit-point [φ,0*î] = [0.6183…,0*î] on the
real axis. Then the angles of the vectors from origin to the points are approximately
equi-angular. Use for convenience the notation fh(h) instead of f°h(1) in the following.

Here the lengthes (abs(fh(h)– φ) ;blue line) and
the angles (arg(fh(h)–
φ)/Pi ;magenta line)
for the points, when the origin is translated to [φ,0*î]:

The magenta line for the angles is nearly
linear. In the following picture the deviation from linearity (the
non-equi-angularity) of the trajectory is shown.

Pic 4:

Where the sinusoidal curve meets the x-axis,
the trajectory(magenta line) meets the y=x-line. We see the slightly distorted deviation
from a straight line which is visually magnified by a rescaling-factor of about
200 and shown with the red sinusoidal curve.